A nice quotation from P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, E. Udovina and D. Vaintrob: Introduction to representation theory. arXiv:0901.0827: «If we needed to make contact with an alien civilization and show them how sophisticated our civilization is, perhaps showing them Dynkin diagrams would be the best choice!»
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Mariano Suárez-Alvarez♦Nov 25 '09 at 12:49

ADE correspond to skew-symmetric cluster algebras of finite type; you get the whole Dynkin (ABCDEFG) classification if you include all finite type cluster algebras (i.e., if you include ones that are skew-symmetrizable and not skew-symmetric).
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Hugh ThomasNov 25 '09 at 16:46

As Mariano said, the ADE Dynkin diagrams classify quivers of finite representation type. But wait, there's more. If you add one more vertex to a Dynkin diagram (in a particular way, not an arbitrary one), you get an extended Dynkin diagram (aka a Euclidian diagram). The extended ADE diagrams classify quivers of tame representation type. This is related to the fact that the extended ADE diagrams give you a positive semi-definite Tits form, while the ordinary ADE diagrams give you a positive definite Tits form.

Let $G$ be a connected graph with the property that all eigenvalues of $G$ lie in $[-2,2]$ (such a $G$ is called cyclotomic). Then $G$ is either one of $\tilde{E}_6,\tilde{E}_7,\tilde{E}_8$, an $\tilde{A}_n$ for $n\ge 2$, a $\tilde{D}_n$ for $n\ge4$, or an induced subgraph of one of these. In other words, the ADE graphs classify the maximal cyclotomic graphs.

I suggest to take a look on a very nice Givental's paper (MR1138519 (92k:58031)):

"Reflection groups in singularity theory."

Here is the review (by V.D. Sedykh): The simple singularities of functions are classified by the Coxeter groups $(A,D,E$-classification). This classification arises in other problems, too (the classifications of the simple Lie algebras, of the finite quaternions groups and so on). The author gives a detailed survey of these results. He also considers the problems connected with the classification of the quasihomogeneous unimodular singularities of functions (the classification of the degenerations of elliptic curves, the theory of automorphic functions and so on) in this paper.